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We show an intrinsic version of Thomason's fixed point theorem. Then wedetermine the local structure of the Hilbert scheme of $\leq 7$ points in$\mathbb{A}^3$. In particular, we show that in these cases the points with thesame extra dimension have the same singularity type. Using these results wecompute the equivariant Hilbert functions at the singularities and verify aconjecture of Zhou on the Euler characteristics of tautological sheaves onHilbert schemes of points on $\mathbb{P}^3$ for $\leq 6$ points.

On singular Hilbert schemes of points: local structures and tautological sheaves